+ apply oa_leq_antisym;
+ [ apply oa_density; intros;
+ apply oa_overlap_sym;
+ unfold binary_join; simplify;
+ apply (. (oa_join_split : ?));
+ exists; [ apply false ]
+ apply oa_overlap_sym;
+ assumption
+ | unfold binary_join; simplify;
+ apply (. (oa_join_sup : ?)); intro;
+ cases i; whd in ⊢ (? ? ? ? ? % ?);
+ [ assumption | apply oa_leq_refl ]]
+qed.
+
+lemma overlap_monotone_left: ∀S:OA.∀p,q,r:S. p ≤ q → p >< r → q >< r.
+ intros;
+ apply (. (leq_to_eq_join : ?)‡#);
+ [ apply f;
+ | skip
+ | apply oa_overlap_sym;
+ unfold binary_join; simplify;
+ apply (. (oa_join_split : ?));
+ exists [ apply true ]
+ apply oa_overlap_sym;
+ assumption; ]
+qed.